in elastomeric polymers
A Thesis by
Stefanie Heyden
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
2015
(Defended October 13, 2014)
To my Dad –
for trying to bring my blood sugar back to non-comatose levels (I promise, I will accomplish a pull-up one day).
And to the boy I came with four years ago – for everything in between.
Acknowledgements
This thesis is the result of four years of work under the continued guidance and inspi- ration of my advisor, Professor Michael Ortiz. His kindness, patience, and unceasing support have served as a constant reminder of how fortunate I was to become a member of his group. Above all, however, I would like to thank him for changing my perspective:
from viewing science as a means of understanding – to getting a glimpse at the beauty behind it all.
Moreover, I would like to express my gratitude to the members of my thesis committee, Professors Guruswami Ravichandran, Kaushik Bhattacharya, and Kerstin Weinberg (Uni- versity of Siegen), who have supported my work through all those years with many kind and truly uplifting words as well as personal advice and guidance, even from afar. In addition, Professor Anna Pandolfi (Politecnico di Milano) has been a wonderful mentor and teacher, whose research visits at Caltech I deeply cherished. Further inspiring bright minds that I was lucky enough to collaborate with include Professors Sergio Conti (Uni- versity of Bonn) and Wolfgang Wall (TUM), whose help and support through theHaus- dorffCenter for MathematicsandInstitute for Advanced Study, respectively, enabled several research stays and scientific adventures outside the United States. Financial support from the Office of Naval Research and the National Science Foundation is also gratefully ac- knowledged.
Looking back on my time at Caltech, I am indebted to many other people who have sur- rounded me over the years. Lydia and Marta, my days as a graduate student were bright- ened every time I stepped into your offices. Cindy, Jonathan, Brandon and Panos – coffee breaks, studying for qualifying exams, or being teaching assistants wouldn’t have been the same without the amazing group of people I could be a part of. My deep appreci- ation also goes to Bo Li as one of those great students whose paths I tried to follow, for passing on his knowledge and introducing me to his computational methods in countless discussions. Aubrie and Jeff, thanks for late-night pizza deliveries to campus through the pouring rain, for converting math variables into farm animals, but most importantly, for your friendship. Gabriela, muchisimas gracias for simply being my best buddy, in- and outside the research bubble.
Finally, I would like to thank my little sister. Her light-hearted spirit always reminds me not to take ourselves too seriously.
Abstract
This thesis aims at a simple one-parameter macroscopic model of distributed damage and fracture of polymers that is amenable to a straightforward and efficient numerical imple- mentation. The failure model is motivated by post-mortem fractographic observations of void nucleation, growth and coalescence in polyurea stretched to failure [Weinberg and Reppel, 2013], and accounts for the specific fracture energy per unit area attendant to rupture of the material.
Furthermore, it is shown that the macroscopic model can be rigorously derived, in the sense of optimal scaling, from a micromechanical model of chain elasticity and failure regularized by means of fractional strain-gradient elasticity. Optimal scaling laws that supply a link between the single parameter of the macroscopic model, namely the critical energy-release rate of the material, and micromechanical parameters pertaining to the elasticity and strength of the polymer chains, and to the strain-gradient elasticity regu- larization, are derived. Based on optimal scaling laws, it is shown how the critical energy- release rate of specific materials can be determined from test data. In addition, the scope and fidelity of the model is demonstrated by means of an example of application, namely Taylor-impact experiments of polyurea rods. Hereby, optimal transportation meshfree approximation schemes using maximum-entropy interpolation functions are employed.
Finally, a different crazing model using full derivatives of the deformation gradient and a core cut-offis presented, along with a numerical non-local regularization model. The nu- merical model takes into account higher-order deformation gradients in a finite element framework. It is shown how the introduction of non-locality into the model stabilizes the effect of strain localization to small volumes in materials undergoing softening. From an investigation of craze formation in the limit of large deformations, convergence studies verifying scaling properties of both local- and non-local energy contributions are pre- sented.
Contents
List of Figures xi
List of Tables xii
Notation xiii
1. Introduction 1
1.1. Elastomeric polymers as structural materials . . . 1
1.1.1. Derivation and usage . . . 1
1.1.2. Limitations and failure . . . 3
1.2. Models of polymer elasticity . . . 6
1.2.1. Macromolecular polymer models . . . 7
1.2.2. From macromolecular to continuum scales . . . 10
1.3. Formulations of non-local damage . . . 19
1.3.1. Motivation . . . 19
1.3.2. Strong and weak non-locality . . . 22
1.3.3. Non-local models of integral type . . . 24
1.3.4. Non-local models of gradient type . . . 27
2. Analytical description of crazing mechanisms 29 2.1. Competing constitutive effects . . . 29
2.2. Local behavior . . . 31
2.3. Growth properties of the deformation-theoretical strain energy density . . 36
2.3.1. Upper bound . . . 36
2.3.2. Lower bound . . . 37
2.3.3. Illustrative examples . . . 38
2.4. Nonlocal regularization . . . 39
3. Optimal scaling and specific fracture energy 42
3.1. Problem formulation . . . 44
3.2. Upper bound . . . 45
3.3. Lower bound . . . 53
3.4. Relation to fracture . . . 53
4. Simulation of Taylor impact experiments 56 4.1. Numerical implementation . . . 56
4.1.1. Optimal transportation meshfree approximation . . . 56
4.1.2. Material failure . . . 61
4.2. Supporting microscopy and experimental calibration . . . 66
4.3. Static and viscoelastic behavior of polyurea . . . 69
4.4. Taylor-anvil tests . . . 72
5. Crazing model using full derivatives and a core cut-off 79 5.1. Problem formulation . . . 79
5.2. Crazing and scaling . . . 82
5.2.1. Some heuristics . . . 83
5.2.2. Local constitutive damage model . . . 86
5.2.3. Numerical non-local regularization model . . . 89
5.2.4. Simulation of craze formation . . . 91
6. Concluding remarks and future work 96 A. Preliminaries 101 A.1. Vector and tensor fundamentals . . . 101
A.1.1. Basis representation and summation convention . . . 101
A.1.2. Vector and tensor operations . . . 102
A.1.3. Tensor analysis . . . 107
A.2. Review of continuum mechanics . . . 109
A.2.1. Finite kinematics . . . 109
A.2.2. Linearized kinematics . . . 113
A.2.3. Rates of deformation . . . 115
A.2.4. Conservation laws . . . 116
List of Figures
1.1. Young’s modulus versus mass density plot for different groups of materials, adapted from [Granta Design, 2014]. . . 2 1.2. Chemical reaction between an isocyanate component and a synthetic resin
blend forming polyurea. . . 3 1.3. Surface profiles (height measured from the deepest point) of a polyurea
specimen tested in uniaxial tension Weinberg and Reppel [2013]. Left:
Initial profile showing initial porosity. Right: Profile after fracture showing proliferation of voids. . . 3 1.4. Crazing process in a steel/polyurea/steel sandwich specimen under open-
ing mode fracture [Yong et al., 2009]. . . 5 1.5. Graphical interpretation of different length measures describing polymeric
structures. . . 9 1.6. Representative volume element used in the eight-chain-model [Arruda and
Boyce, 1993]. . . 13 1.7. Network of cross-linked polymer chains upon deformation (adapted from
Weiner [2002]). . . 18 2.1. a) Deformation-theoretical strain energy density Wloc(F) for an isotropic
polymer deforming in uniaxial extension. The strong and weak chains both fail at a critical stretch λc = 2 and have bond-binding energies of kEb
BT = 2 and kEb
BT = 0.2, respectively. b) Schematic of functions with linear, sublinear and superlinear growth (reproduced from [Heyden et al., 2014]) . . . 39 3.1. Schematic of the crazing construction showing a slab divided into unit cells
under prescribed opening displacementsδ. . . 46
3.2. Schematic of the deformation mapping (as shown for one periodic unit cell) used in the upper bound construction. . . 47 3.3. Schematic of the first mappingf used in the upper bound construction. . . 47 3.4. Schematic of the second mappinggused in the upper bound construction. 49 4.1. Spatial discretization used in the optimal transportation meshfree approxi-
mation schemes (adapted from [Li et al., 2010]). Material points are shown in red, whereas nodal points are shown in white. An exemplary circular local neighborhood of nodal points is shown at timetk. . . 59 4.2. Time discrete Lagrangian dynamics (adapted from [Li, 2009]). . . 61 4.3. Left: Advancing crack showing a zoom of the crack front propagating in
the direction of crack front velocity v (adapted from [Pandolfi and Ortiz, 2012]). C(t) and C(t+∆t) are the original and extended crack set, respec- tively. Right: Set of eroded material points forming a crack and respective -neighborhood. . . 64 4.4. Visualization of the material point eigenerosion approach (adapted from
[Pandolfi et al., 2014]). Black dots denote members of the crack set ({∗ , 0}), whereas gray dots belong to the -neighborhoods of failed material points ({∗ , 0}). The thickness of the -neighborhood is 2 and, after crack propagation, the increment in the crack’s-neighborhood is∆Vp. . . 66 4.5. Uniaxial tension test of polyurea 1000 with feed of 2mm/s [Reppel et al.,
2012]. Left: Thin-strip specimen stretched quasistatically up to failure.
Right: Determination of the elastic strain energy density by data reduction. 67 4.6. Micrograph of the fracture surface of a polyurea specimen after failure
in uniaxial tension (adapted from [Reppel et al., 2012]), whereby shaded areas represent voids. . . 68 4.7. Comparison of different elastic material models shown in (b) fitted to low-
strain rate experimental data of Sarva et al. [2007] at ˙ = 0.0016s−1 as de- picted in (a). . . 70
4.8. Left: Cauchy stressσ versus true strain = logλfor polyurea at different true strain rates ˙; data collected from Roland et al. [2007], Sarva et al.
[2007], Yi et al. [2006], Zhao et al. [2007], Amirkhizi et al. [2006] and Rep- pel et al. [2012]. Right: Results for fittings of a Prony series formulation as introduced in Section 4.3 to the experimental data set. . . 72 4.9. Taylor-anvil test of polyurea 1000 rod; experiments have been performed
by Mock and Drotar [2006] at NSWC.R0= 6.29603 mm,L0= 25.7353 mm, andv= 245 m/s. . . 73 4.10. Snapshots of the simulated deformation atv= 245 m/s impact velocity. . . 75 4.11. Snapshots of the simulated deformation atv= 332 m/s impact velocity. . . 75 4.12. Snapshots of the simulated deformation atv= 424 m/s impact velocity. . . 76 4.13. Comparison of the recovered target after shot between experiments and
simulations with impact speeds v = 245m/s (a), v = 332m/s (b) and v = 424m/s (c). . . 77 4.14. a) Normalized specimen height versus time at impact speedsv= 245 m/s,
v = 332 m/s and v = 424 m/s. b) Logarithmic convergence plot showing total accumulated specimen length over time for different mesh sizes. . . . 78 5.1. Schematic of the topology of fibril nucleation and growth. Left: Planar
network of cylindrical cavities that provide the nucleation sites for fibrils.
The inscribed circuit cannot be reduced continuously to a point, which il- lustrates the topological transition undergone by the body as a result of nu- cleation. Right: Distribution of fibrils resulting from the expansion of the nucleation sites under transverse uniaxial deformation. Since the deforma- tion after nucleation is continuous outside the boreholes, the topology of the body does not change upon deformation. In particular, the structure of irreducible circuits such as inscribed remains unchanged. . . 80 5.2. Expansion of borehole in a concentric incompressible cylinder. . . 82 5.3. Infinite slab of thickness 2H subject to prescribed opening displacements
δon its surface. . . 82
5.4. Assumed deformation pattern describing the process of crazing. The defor- mation is assumed to localize to a thin layer of thicknessaand, elsewhere, the slab is assumed to undergo a rigid translation through the prescribed opening displacement ±δ. The layer is then further subdivided into iden- tical cubes of sizea, each of which undergoes the deformation illustrated above. The void region arises from the equator of the cube as shown on the right. . . 83 5.5. Subset of cubature points on the unit sphere for which fiber orientationζi
under uniaxial loading in the direction of λ lies in a purely compressive zone (and hence will not fail in the limit of large deformations). . . 87 5.6. Cubature points on the unit sphere before (left) and after rotation (right). . 87 5.7. Examples of different sets of interface energy contributions between tetra-
hedral element 1 and its neighbors. . . 90 5.8. a) Schematic of the domain of analysis spanning one quarter of the cubic
periodic unit cell. b) Initial tetrahedral 8,766-element mesh used in calcu- lations. . . 92 5.9. Sequence of equilibrium configurations under increasing prescribed open-
ing displacement. Superimposed on the figure are level contours of volu- metric deformation det(F). . . 93 5.10. a) Normalized local energy Wloc/W∞a3 vs. normalized opening displace-
ment δ/a. b) Normalized nonlocal energy Wnon/W∞a2` vs. normalized opening displacementδ/a. . . 94 5.11. Dependence of the normalized nonlocal energy Enon/W∞a2` on the nor-
malized core cutoffradiusb/afor fixedδandaandδa. . . 94 A.1. Deformation mappingϕ between a body in the reference configurationΩ
and its deformed configurationϕ(Ω). . . 110
List of Tables
4.1. Elastic material parameters of a Neo-Hookean solid for polyurea used in OTM-simulations of Taylor-impact experiments. . . 69 4.2. Moduli and relaxation times for polyurea (units are [MPa] and [s]) obtained
from fitting [Knauss and Zhao, 2007]. . . 71 4.3. Specifics of the three different Taylor-anvil test cases. . . 73 A.1. Deformation measures in linearized kinematics and corresponding defor-
mation rates. . . 115 A.2. Overview of work-conjugate pairs. . . 119 A.3. Summary of stress measures. . . 119
Notation
The following table gives an overview of different symbols used in the present work.
In general, scalar quantities are denoted by lowercase letters, whereas bold lower- and uppercase letters denote vector and tensor quantities, respectively.
A, a area in the undeformed and deformed configuration
x,X coordinates in the deformed and undeformed configuration x, y, z Cartesian coordinates
C right Cauchy-Green deformation tensor
ei unit vectors in Cartesian coordinates
E Young’s modulus
F deformation gradient tensor
f vector of body or surface forces
fint,fext vector of internal and external forces
J Jacobian
L velocity gradient tensor
n unit normal vector
Nei shape functions
Q heat
S entropy
t time
∆t time increment
t traction vector
T temperature
u displacement vector
v velocity vector
L Langevin function
kB Boltzmann constant
lt total polymer chain length
lm monomer length
β reciprocal absolute temperature
σ variance
λr relative stretch
ν(N) distribution of polymer chains withN monomers
ρ(lt, N) probability distribution of chains withN monomers of total lengthlt κ,µ,λ bulk modulus, shear modulus, Lam´e parameter
H Hamiltonian of a system
Z partition function
L stochastic point process
T Delaunay tesselation
I second-order identity tensor
E Green-Lagrange strain tensor
linearized strain tensor
P first Piola-Kirchhoffstress tensor
σ Cauchy stress tensor
Ndim number of spatial dimensions
α∞ attenuation function
D(ζ) damage distribution function
Eb energetic cost of polymer chain failure kl/ku lower/upper constants
l intrinsic length
Φ specific energy per unit area
Gc critical energy release rate
1. Introduction
This section gives an overview of both application fields and current limitations of elas- tomeric polymers and summarizes the main modeling approaches that have been put forth in the literature. Furthermore, formulations of non-local damage in the framework of elasticity are discussed.
1.1. Elastomeric polymers as structural materials
Elastomeric polymers have recently been identified as suitable structural materials in a wide range of applications, including shock mitigation as well as blast protection [El Sayed et al., 2009]. They have further been identified as promising in transparent armor appli- cations, in which specific “ports” (transparent to visible and other wavelengths) need protection against foreign object impact [Albrecht et al., 2012]. In the following, details of the derivation and usage of one such elastomeric polymer, viz. polyurea, will be dis- cussed. In addition, material failure characteristics and resulting current limitations will be reviewed. These form a basis for subsequent material modeling approaches.
1.1.1. Derivation and usage
Elastomeric polymers mark a group of materials with a long-standing tradition in the field of soft materials. Prime examples of their main characteristics include high damp- ing capabilities as well as high stiffness-to-weight ratio. Moreover, lightweight mono-
10000 100
10 1000
Density [kg/m ]3
Young’s modulus [GPa]
0.001 1 1000
Elastomers Plastics
Composites Metals and alloys Natural materials
Technical ceramics
Foams
Figure 1.1.: Young’s modulus versus mass density plot for different groups of materials, adapted from [Granta Design, 2014].
lithic coatings made of elastomeric polymers have also shown excellent mitigation prop- erties in the event of powerful explosions, as well as the capability of retaining structural fragments produced by blast impact. They can easily be applied via spray-on or cast-on techniques, and different types of reinforcement and laminated designs have been inves- tigated by several researchers [Colakoglu et al., 2007; Grujicic et al., 2006; Wambua et al., 2007]. A solution methodology for projectile impact on such structures was developed based on contact load duration, through-thickness and lateral transit times in [Lin and Fatt, 2006]. From their typical application as a protective coating of concrete and steel structures, it follows that such elastomeric coatings have to withstand rapid loadings such as during impact, collisions or explosions up to total failure of the material. An example of an elastomeric polymer used as a structural material is polyurea, an elastomer that is derived from the chemical reaction of an isocyanate component and a synthetic resin blend. Polyurea has been shown to exhibit beneficial properties for shock mitigation.
In particular, it is characterized by a high strain rate sensitivity, large maximum defor- mations and good adhesion properties to many materials. These characteristics make it suitable for protective coatings on structures and have motivated its experimental char- acterization [Chakkarapani et al., 2006; Jiao et al., 2006, 2007, 2009; Knauss and Zhao,
OCN R
NCO H N2
R’
NH2
+ R
N
H N
H
R’
N
H N
H O
n
Diisocyanate Polyamine Urea
Figure 1.2.: Chemical reaction between an isocyanate component and a synthetic resin blend forming polyurea.
Figure 1.3.: Surface profiles (height measured from the deepest point) of a polyurea spec- imen tested in uniaxial tension Weinberg and Reppel [2013]. Left: Initial profile showing initial porosity. Right: Profile after fracture showing prolif- eration of voids.
2007; Roland and Casalini, 2007; Roland et al., 2007; Sarva et al., 2007].
With a quasistatic elastic modulus of about 70MPa [Knauss and Zhao, 2007], polyurea lies in the range of elastomers shown in Figure 1.1. However, as investigated in [Jiao et al., 2006, 2007, 2009], a ring-up in pressure due to wave propagation and reflection in thin material samples under pressure shear-plate impact loading greatly increases the material’s strength.
1.1.2. Limitations and failure
The use of polymers as structural materials is critically limited by their tendency to de- grade by distributed damage or to fail by fracture, sometimes in a brittle manner (cf., e. g., Andrews [1968]; Argon [2013]; Bikales [1971]; Grellmann and Seidler [2001]; Kausch
[1985]; Kinloch and Young [1983]; Williams [1984] for reviews). Damage in polymers de- formed under tensile loading often takes the form of distributed voids [Jiao et al., 2006, 2007, 2009; Weinberg and Reppel, 2013], cf. Fig. 1.3. Voids nucleate heterogeneously from flaws or inclusions, Fig. 1.3a, and subsequently grow under tension, Fig. 1.3b, re- sulting in softening or loss of bearing capacity of the material (cf., e. g., Cho and Gent [1988]; Gent [1973]; Gent and Wang [1991]). Likewise, fracture in polymers can often be traced to the formation of crazes (cf., e. g., Donald and Kramer [1982]; Henkee and Kramer [1986]; Kausch [1983]; Kramer and Berger [1990]; Sanderson and Pasch [2004]), Fig. 1.4.
Crazes are thin layers of highly localized tensile deformation. Craze surfaces are bridged by numerous fine fibrils, themselves consisting of highly oriented chains, separated by connected voids. Crazes undergo several stages along their formation, including nucle- ation, growth and final breakdown, resulting in the formation of a traction-free crack, or fracture. Craze initiation is likely the result of heterogeneous cavitation at flaws loaded under conditions of high triaxiality. Craze propagation has been linked to a meniscus instability resulting in the formation of fibrils. This analogy is immediately suggestive of some role played by surface energy or other similar physical properties not accounted for by bulk behavior. Eventually, crazes break down to form cracks. Experimentally, crazes are easily identified and observed fractographically by a variety of techniques including optical interferometry, light reflectometry, dark-field electron microscopy, and others.
Owing to its engineering importance, polymer damage and fracture have been the fo- cus of extensive modeling. A number of micromechanical and computational models, ranging from atomistic to continuum, have been put forth (cf., e. g., Baljon and Rob- bins [2001]; Basu et al. [2005]; Drozdov [2001]; Estevez et al. [2000a,b]; Krupenkin and Fredrickson [1999a,b]; Leonov and Brown [1991]; Reina et al. [2013]; Rottler and Rob- bins [2003, 2004]; Saad-Gouider et al. [2006]; Seelig and Van der Giessen [2009]; Socrate et al. [2001]; Tijssens and van der Giessen [2002]; Tijssens et al. [2000a,b]; Zairi et al.
Figure 1.4.: Crazing process in a steel/polyurea/steel sandwich specimen under opening mode fracture [Yong et al., 2009].
[2008]). These models include consideration of nucleation and growth of voids, craze nucleation, network hardening and disentanglement, chain strength, surface energy and others, which account, to varying degrees, for the observational evidence and relate macro- scopic properties to material structure and behavior at the microscale. In parallel, a large mathematical literature has evolved, discussing the possibility of cavitation in local mod- els and nonlocal extensions which may ensure existence of minimizers; see, for example, Ball [1982]; Conti and DeLellis [2003]; Henao and Mora-Corral [2010]; James and Spector [1991]; M ¨uller and Spector [1995].
Despite these advances, the connection between micromechanical properties and poly- mer fracture, and specifically any scaling laws thereof, has defied rigorous analytical treatment and characterization. Of special interest is the identification of optimal scal- ing laws relating the macroscopic behavior to micromechanical and loading parameters.
Such optimal scaling laws are established by producing upper and lower bounds of a power-law type with matching exponents for all parameters in both bounds. Optimal scaling methods were pioneered by Kohn and M ¨uller [1992] as part of their seminal work on branched structures in martensite, and have since been successfully applied to a num- ber of related problems, including shape-memory alloys, micromagnetics, crystal plastic- ity, and others [Choksi et al., 1999; Conti, 2000; Conti and Ortiz, 2005; Kohn and M ¨uller, 1992, 1994].
Fokoua et al. [2014a,b] have recently applied those analysis tools to ductile fracture of metals. They specifically considered the deformation, ultimately leading to fracture, of a slab of finite thickness subject to monotonically-increasing normal opening displace- ments on its surfaces. In addition, they posited two competing constitutive properties, namely sublinear energy growth and strain-gradient hardening. Sublinear growth (for comparison, the energy of linear elasticity exhibits quadratic growth) is a reflection of the work-hardening characteristics of conventional metallic specimens, and gives rise to well-known geometric instabilities such as the necking of bars, sheet necking, strain lo- calization and others (cf., e. g., McClintock and Argon [1966]). In metals undergoing ductile fracture, this inherently unstable behavior is held in check by a second fundamen- tal property of metals, namely strain-gradient hardening [Fleck and Hutchinson, 1993, 1997, 2001; Fleck et al., 1994]. Under these assumptions, Fokoua et al. [2014a,b] showed, through rigorous mathematical proofs, that ductile fracture emerges as the net outcome of two competing effects: while the sublinear growth of the energy in the large-body limit promotes localization of deformation to failure planes, strain-gradient plasticity stabilizes this localization process in its advanced stages, thus resulting in a well-defined specific fracture energy.
1.2. Models of polymer elasticity
The mathematical description of polymeric materials commonly occurs on two different scales. On the one hand, the framework of continuum mechanics (see Section A.2 for a brief review) is used to describe hyperelastic materials in the finite deformation range;
on the other hand, an alternative approach derives from statistical mechanics by describ- ing individual polymer chains. The latter approach for modeling polymers offers the advantage of being able to derive the free energy of an individual polymer chain from first principles only, whereas hyperelastic material models are more phenomenological in nature Gloria et al. [2013].
However, bridging these scales and relating energy densities in the continuum descrip- tion to the energy of a network of polymer chains bears several difficulties. First and fore- most, assumptions on the interactions between different chains are necessary to facilitate the analysis and pass from individual polymer chains to a network of chains. Further- more, the way in which the total energy decomposes into the different contributions from the lower scales must be specified.
1.2.1. Macromolecular polymer models
Derived from their macromolecular structure, polymers can be modeled as ensembles of chains with energyf(λ,T), where λis the stretch of the relative position vector between the chain ends andT is the absolute temperature. Prime examples of both uncorrelated and correlated chain models are the freely jointed chain model (FJC) in a microcanon- ical ensemble formulation for Gaussian and non-Gaussian statistical approximations as well as the Kratky-Porod and wormlike chain (WLC) models as representatives of stiffer polymer models, see e.g. [Weiner, 2002].
The freely jointed chain model
The freely jointed chain model approximates a polymeric structure of total lengthltas an ideal chain of monomers possessing equal lengthslmand directions that are uncorrelated to the neighboring ones, cf. Figure 1.5.
Based on the large number of monomers N in a polymer, the probability density of the relative endpoint position vector approaches a Gaussian distribution, which in three di- mensions results in
pFJC(le) = p0 (
√
2πσ)3e−
l2 e
2σ2 (1.1)
with varianceσ =p
N lm2/3 and normalizing constantp0. The total free energy of a single chain then follows as
fFJC=−kBTlnΩ=fFJC0 +3
2kBT N λ2r, (1.2)
wherefFJC0 denotes the free energy of the undisturbed freely jointed chain,kB andT are the Boltzmann constant and absolute temperature, respectively,Ωstands for the proba- bility of obtaining a certain endpoint position andλr is the relative stretch of the chain as the end-to-end length le divided by lt. For a graphical interpretation of the differ- ent lengths describing the polymeric structure, please refer to Figure 1.5. From Equa- tion (1.2), the force-stretch relation of a single polymer chain can be calculated as
FFJC=3kBT
lm λr. (1.3)
This expression, however, is only valid in the small strain regime due to the Gaussian ap- proximation with regard to changes in entropy [Weiner, 2002]. Following a non-Gaussian statistical approach (Kuhn and Gr ¨un [1942]) with probability
ΩFJC(le) =Ω0exp −N λrL−1+ ln L−1 sinh(L−1)
!!
(1.4)
and recalling thatλr =le/lt, the free energy of a single chain follows as
fFJC=fFJC0 +kBT N(λrL−1+ lnL−1sinh(L−1)−1). (1.5)
Furthermore, the force-stretch relation of a polymer chain, which is not restricted to the small strain regime, now specifies to
FFJC= 1
βlmL−1(λr). (1.6)
lt le=lt le
le
lt
lm lm
lm
l l
p«
tl
p»l
tl l
p»
tFigure 1.5.: Graphical interpretation of different length measures describing polymeric structures.
In these expressions, β is the reciprocal thermodynamic temperature according to β = 1/kBT, andL−1denotes the inverse Langevin function with
L(λr) = coth(λr)−λr . (1.7)
The inverse of Equation (1.7) can be restated using a Pad´e approximation [Miehe et al., 2004] resulting in
L−1= 3−λ2r 1−λ2r
λr , (1.8)
and it thus can be seen that in the small strain limit, the force-stretch relation based on Gaussian and non-Gaussian statistics coincide.
The wormlike chain model
The freely jointed chain model oversimplifies the macromolecular structure, resulting, e.g., in a non-existing stiffness against bond bending. Consequently, more elaborate mod- els have been developed, which correlate the different chain segments.
The Kratky-Porod model for example introduces an energy dependency on the bonding angle, and thus penalizes chain rotations [Cross, 2006]. The free energy and force-stretch
relation of a single chain derived in this way take the forms, respectively,
fWLC = fWLC0 +kBT lt lp
1 2λ2r +1
4 1 1−λr −1
4λr
!
and (1.9)
FWLC ≈ kBT
lp λr+1 4
1
(1−λr)2 −1 4
!
, (1.10)
withlp denoting the persistence length as a measure of the stiffness of the tangled poly- mer chains.
1.2.2. From macromolecular to continuum scales
Approaches for bridging the scales between single polymer chains and cross-linked chain networks forming a continuum mainly differ by the set of assumptions on which they rely.
A first and common assumption is that the sum over all free energies of the individual chains gives the total free energy of the polymer network [Gloria et al., 2013]. Further assumptions concern the specification of how isolated chains interact with each other, and different models are summarized below.
The Treloar model
First introduced by Treloar [1949], single chains are assumed to move in an affine manner according to the global deformation gradient (also known as the affine assumption or Cauchy-Born rule). The total strain energy density of the polymer network then follows as
WTreloar(F) = Z
R+
Z
R+
Z
S2
Wchain
lt, λζ, N
dσ(ζ)dρ(lt, N)dν(N), (1.11)
whereν(N) specifies the distribution of polymer chains consisting ofN monomers, and ρ(lt, N) describes the probability distribution of chains consisting ofN monomers having lengthlt in the undeformed configuration. Furthermore,S2 denotes the unit sphere and ζ∈S2.
Simplifications to this model can be introduced by assuming that the number of monomers of each polymer chain is constant and that the total chain length in the undeformed con- figuration is given by lt =
√
N lm [Flory, 1969]. Under these conditions, the total strain energy density of the network may be written as
WTreloar(F) = Z
S2
Wchain√
N lm, λζ
dσ(ζ). (1.12)
The Arruda-Boyce model
In the large deformation limit, the Treloar model overestimates the energy stored in the material upon deformation. In order to remedy this deficiency, different ways of relaxing the affine assumption have been proposed in the literature. One of these models was in- troduced by Arruda and Boyce [1993], which relaxes the affine assumption by evaluating a representative volume element and its geometric response. Thereby, the representative volume element is a cube consisting of eight individual polymer chains originating from its center and connecting to each of its corners (see Figure 1.6 for reference).
The model rests upon the assumption that the representative volume element aligns it- self based on the principal directions of the macroscopic deformation gradient and de- forms according to its principal stretches (whereby no repulsion between polymer chains is taken into account) [Gloria et al., 2013]. Following this assumption, the total strain energy density is proportional to the energy of individual chains in the deformed rep- resentative volume element. By noting that the deformation ratio of each chain may be
calculated as λchain
lt = s
λ21+λ22+λ23
3 =
rI1
3 (1.13)
with total chain lengthlt=
√
3/2 in the undeformed configuration, the energy density of the polymer network (assuming isochoric deformations) may be written as
WArruda-Boyce(F) =n βN
rI1
3
√ N l N l L−1
rI1
3
√ N l N l
(1.14)
+ log
L−1
qI1 3
√ N l N l
sinhL−1 qI1
3
√ N l N l
. (1.15)
Here,ndenotes the chain density and, as introduced above,βstands for the reciprocal ab- solute temperature. In order to expand this energy and include volumetric effects, strain energies modeling the volumetric material response upon deformation may be added, such as the well-known Helmholtz volumetric energy [Weiner, 2002]
WHelmholtz(F) =κ
J2−1−2 log(J)
. (1.16)
The variational model
A different way of relaxing the affine assumption lies in the introduction of a minimiza- tion principle (also known as the variational model) [Gloria et al., 2013], which rests upon the idea that the total free energy of the polymer is minimized by the positions of cross-linking points when the system reaches equilibrium. Starting from a macroscopic sample Ω consisting of a network of cross-linked polymer chains (whereby cross-links are assumed permanent and entanglements of chains are neglected), the Hamiltonian of
φ λ1
λ2 λ3
Figure 1.6.: Representative volume element used in the eight-chain-model [Arruda and Boyce, 1993].
the system follows as
H(u, s) =Hvol(u, s) +X
i
Hi(u, si). (1.17)
Here,u denotes the positions of cross-links, andsi are the positions of monomers associ- ated with chaini. Furthermore, the additive decomposition of the Hamiltonian enables a split into volumetric contributions Hvol and individual chain contributionsHi. The free energy of the variational model can then be stated as
A(F) =−1
βln(Z), (1.18)
withZbeing the partition function according to
Z= Z
U
Z
S1(u)
Z
S2(u)
...
Z
Sn(u)
e−βH(u,s)dsn. . . ds2ds1du. (1.19)
In this expression, U and Si(u) are the sets of admissible positions of cross-links and monomers, respectively. Further simplifications introduced in the model are a restriction of chain interactions via cross-links only, as well as the assumption that monomer po- sitions si are decoupled. These simplifications lead to a coarse-grained model that only
depends on cross-link positionsu:
A(F) =−1 βln
Z
U
exp−βHF(u,β)du
!
, (1.20)
where
HF(u, β) =Hvol(u)−X
i
1 βln
Z
Si(u)
exp(βHi(u, si))dsi
!
. (1.21)
The Treloar assumption of a polymer network deforming affinely according to the macro- scopic deformation gradient may be interpreted in this context as restricting the integral R
U in Equation (1.20) to evaluations atu(x) =F·xonly. In the variational model, however, the affine assumption only restricts the admissible set of cross-link positions U on the boundary, whereas the minimum of the coarse-grained Hamiltonian A(F)'infuHF(u, β) (withΩ→ ∞in the thermodynamic limit) gives the free energy in the interior.
In order to further specifyHF(u, β), the notion of a discrete network is introduced, and its main features and results for the variational model are summarized here (a more detailed description can be found in Gloria et al. [2013]). With a stochastic point processL inR3 as a sequence of random points inR3, and a Delaunay tesselationT ofLinR3 specified by the tetrahedral mesh associated withL, a scaling according to
L=L and T=T (1.22)
can be introduced. The energy associated with a deformation fieldu∈ S(T) then follows as
A(u, D) =3X
e∈E
Wnn |e1−e2|,|u(e1)−u(e2)| |e1−e2|
!
(1.23)
+X
T∈T
|T|Wvol(det∇u), (1.24)
wheree= (e1, e2),WnnandWvoldenote the edge between vertices, energy of the deformed edges and volumetric energy, respectively.
Now, only Wnn remains to be specified. By introducingNe as the number of monomers per edge, and after some simplifications, the energy of the deformed edges follows as
Wnn(|e|, λ) = n βNe
√λ
NeL−1 λ
√ Ne
! + log
L−1 √λ
Ne
sinhL−1 √λ
Ne
, (1.25)
which completes the model.
The B ¨ol-Reese model
A similar approach to the variational model may be found in [B¨ol and Reese, 2005], in which both volumetric and polymer chain energies are considered. In this model, a tetra- hedral mesh of a macroscopic sampleΩis generated.
Subsequently, the volumetric material response is associated with each tetrahedron of the mesh, whereas the edges of each element represent polymer bundles and thus introduce discrete energies associated with individual polymer chains.
For simplicity, the energies of polymer bundles are taken to be multiples of the energy of a single polymer chain. These energies can thus be written as a function Wedge(λedge) = Wedge
l
edge
Ledge
, wherebyLedge andledgedenote edge lengths in the reference and deformed configurations, respectively.
With decreasing mesh size, the B¨ol-Reese model converges to a continuum model. How- ever, it is important to note that the resulting model highly depends on the details of the tetrahedral mesh, as discussed further in B¨ol and Reese [2005].
Network theory of entropic elasticity
In a different framework originating from statistical mechanics, a standard description to go from single polymer chains to a network description is purely based on entropic contributions. Here, an amorphous network of cross-linked long-chain molecules is con- sidered, and the undeformed configuration is given by a unit volume comprising a total ofν= 1, ..., N cross-linked long-chain molecules, with respective numbers of linksnν and link lengthsb. Under the assumptions of sufficiently long chains which are far away from the fully extended limit, as well as a motion of cross-linking points according to superim- posed deformations (neglecting thermal motion), the change in entropy of the amorphous network upon deformation can be written as [Weiner, 2002]
∆Sν =− 3kB 2nνb2
r2(ν)−R2(ν)
. (1.26)
In this expression,R(ν) andr(ν) denote the end-to-end vector of theνth chain before and after deformation, respectively (see Figure 1.7 for reference), and R and r denote their lengths, respectively. By further assuming the same affine transformation described in terms of the strain tensorEIJ = 12(CIJ−IIJ) for all cross-linking points (which is the affine assumption used in Treloar’s model) and using the relationr(ν)2−R(ν)2= 2EIJRI(ν)RJ(ν), the total entropy of the system evaluates to
S(EIJ) =−3kEIJ b2
XN ν=1
RI(ν)RJ(ν)
nν , (1.27)
where the entropy of the undeformed network is taken as a reference point. A material tensor describing the undeformed configuration may be introduced as
KIJ = 3 b2
XN ν=1
RI(ν)RJ(ν) nν
, (1.28)
which in the case of an homogeneous and isotropic undeformed network results inKIJ = KδIJ with
K = 1 b2
XN
ν=1
R2(ν)
nν . (1.29)
For bothEIJ andKIJ, the subscripts in uppercase letters refer to components of tensors in the reference configuration (as opposed to lowercase letters, which denote the deformed configuration as further described in Section A.2). Details regarding the evaluation ofK and its dependence on the cross-linking process may be found in [Weiner, 2002], and the main results and assumptions are summarized below. Based on the observation that K describes the undeformed network of polymer chains, it can be seen that its value can only be specified further by details describing the cross-linking process. For simplicity, all cross-links between polymer chains are assumed to occur simultaneously and form at adjacent points of different chains. The total ofN chains is furthermore subdivided into m groups, whereby each group consists ofcα chains havingnα links (α = 1, ..., m). With an end-to-end chain distance expressed byR(p, α) withp= 1, ..., cα, it follows that
K = 1 b2
Xm α=1
1 nα
cα
X
p=1
R2(p, α). (1.30)
Assuming a random cross-linking process,R2(p, α) may be calculated via a Gaussian dis- tribution with variancenαb2so that
cα
X
p=1
R2(p, α) =cαnαb2, (1.31)
and thusK =N. Using these simplifications, Equation (1.27) may be restated as
S(EIJ) =−kBN ELL, (1.32)
φ
R(ν) r(ν)
Figure 1.7.: Network of cross-linked polymer chains upon deformation (adapted from Weiner [2002]).
which can be reformulated in principal stretches as
S(λ1, λ2, λ3) =−kBN 2
X3 L=1
(λ2L−1). (1.33)
Finally, the Helmholtz free energy F of the full unit volume network subject to an arbi- trary deformation requires the addition of an energy componentU(v, T), which depends on volumetric changes. The free energy of the amorphous network then follows as
F(λ1, λ2, λ3, T) =U(v, T) +kBT N 2
X3 L=1
(λ2L−1), with v=λ1λ2λ3. (1.34)
In strain energy density form, and with an exemplary volumetric contribution ˆW(J, T) added, we arrive at
W(F, T) = ˆW(J, T) +kT
2 KIJEIJ = ˆW(J, T) +kT N
2 δIJEIJ. (1.35)
This energy density is the basic representation of a Neo-Hookeansolid, and it is further- more the three-dimensional extension of Equation (1.2). Here and in the following, F denotes the deformation gradient tensor, and the Jacobian J = detF represents the rela- tive volumetric change.
1.3. Formulations of non-local damage
A number of non-local regularization models have been put forth in the literature in an attempt to overcome ill-posed boundary value problems arising in continuum damage models. Due to the presence of softening in these models, the governing field equations loose ellipticity, and a unique solution to the resulting algebraic system does not exist (as shown analytically for the case of wave propagation in a strain-softening bar in [Baˇzant and Belytschko, 1985]). As a result, deformations are observed to localize in narrow bands, with band widths restricted by the spatial discretization size.
Solution approaches to stabilize this process introduce an internal material length scale as well as non-local terms. Two main strategies can be distinguished in the introduction of non-local terms, which are of integral and gradient type. Prime examples of both approaches will be reviewed in this section. For a comprehensive review, see, e.g., [Baˇzant and Jir´asek, 2002], which forms the basis for the following brief review.
1.3.1. Motivation
Nonpolar materials as discussed in [Noll, 1972] constitute a class of materials, for which the stress value at a given point depends on the deformation and temperature evaluated at this point only (and, in some cases, also the history of deformation). The underlying assumption that the material can be treated as a continuum even at an arbitrarily small scale implies the possibility of decomposing a finite body into infinitesimal material vol- umes whose interactions are restricted to the level of balance equations. This assumption, however, is an idealization, and neglects any internal material structure or microstruc- tural details. Microstructural details may be described by spatial variations of material properties, yet their size range over different orders of magnitude renders this approach expensive in practical applications. More importantly, the continuum assumption breaks down at smaller scales and is hence no longer applicable . Therefore, it is important to
choose a resolution level below which the details of the internal structure are only indi- rectly taken into account as effective material properties. This continuum assumption is justified if the characteristic wave length of the deformation field does not lie below the material model’s resolution level. For static applications, the characteristic wave length may be viewed as the minimum region size into which strain is able to localize. One way of avoiding the need for resolution refinement in the case of characteristic wave lengths below resolution level is to introduce generalized continuum formulations.
The first generalized continuum formulation can be found in [Cosserat and Cosserat, 1909], in which material particles have not only translational but also rotational de- grees of freedom. These additional degrees of freedom are defined by rotations of a rigid frame of mutually orthogonal unit vectors. In the time that followed, generaliza- tions of Cosserat’s original theory were developed using additional fields independent of the displacement field. An example can be found in the continuum with microstructure [Mindlin, 1974], in which a microscopic deformation gradient is introduced (which gives, in the special case of orthogonal tensors, the previously described Cosserat micropolar continuum).
A different group of enriched continua (also known as higher-grade materials or gradient theories) is formed by constitutive models incorporating gradients of strain, thus keep- ing the displacement field as the only independent kinematic field. First, gradients of rotations were considered, which are the strain gradient components corresponding to curvature (see, e.g., [Grioli, 1960]). Afterwards, gradients of stretch were included into the theory [Toupin, 1962], as well as higher-order gradients [Green and Rivlin, 1964].
In addition to deviations from local constitutive models at small scales, which are based on microstructural heterogeneities on the characteristic length scale, different motiva- tions of non-locality were proposed in the 1970s. The second main motivation was the strain-softening character of distributed damage. In the case of a local inelastic consti- tutive law with strain-softening damage, numerical as well as analytical results showed
localization of damage into a zone of zero volume [Baˇzant, 1976]. As a result, the nu- merical solution becomes unobjective with regard to the chosen mesh and converges, for decreasing mesh size, to a solution characterized by zero energy dissipation during fail- ure. Two different reasonings can be given for this behavior, and for a one-dimensional dynamic problem, these reasons can be explained as follows:
• In the case of negative tangential stiffness, materials are characterized by an imagi- nary wave speed, and thus lose (apart from special cases of materials) the ability to propagate waves.
• The initial boundary value problem becomes ill-posed, and transitions from a hy- perbolic to an elliptic type. Owing to this change in type, finite changes in the dynamic solution can result from infinitesimal changes in the initial conditions.
For multidimensional tangential stiffness tensors lacking positive definiteness, materi- als can still possess some real wave speeds and therefore propagate stress waves. The direction of stress propagation however is no longer arbitrary, resulting in ill-posed ini- tial boundary value problems that are not necessarily of elliptic type. It has been shown in later investigations (please refer to [Baˇzant, 1976, 1984; Baˇzant and Cedolin, 1979;
Baˇzant and Oh, 1983; Baˇzant et al., 1984; Cedolin and Baˇzant, 1980; Pietruszczak and Mr `oz, 1981] for details) that by introducing a characteristic length in order to model non-local strain softening behavior, the localization of damage can be prevented by regu- larizing the boundary value problem and making it well-posed. As a result, convergence to physically meaningful solutions is achieved.
A final motivation for the introduction of non-locality into constitutive theories is given by size effects. Here, the term size effects denotes the dependence of nominal strength on structure size. For purely local material behavior independent of a characteristic ma- terial length, size effects may be described by power laws (e.g. in linear elastic fracture mechanics). In case of non-locality on the other hand, size effects are of transitional
type, and different power laws are needed in order to describe size effects at scales much smaller/larger than the characteristic length.
The different motivations leading to an introduction of non-local behavior into constitu- tive theories can hence be summarized as follows:
• material heterogeneities leading to small-scale deviations from local continuum models,
• ill-posed initial boundary value problems, which arise in strain-softening damage formulations and lead to unobjective numerical solutions,
• size effects observed in experiments and discrete simulations.
1.3.2. Strong and weak non-locality
Two main strategies can be distinguished in the field of non-local regularization models, which are of integral and gradient type. The former category describes models in which non-local terms are included by introducing weighted averages of local internal variables, whereby averaging is performed over a set of neighboring points close to the point under consideration. In gradient-type approaches, on the other hand, the introduction of non- locality relies on higher-order gradients of non-local variables. Differential equations then describe the evolution of control variables, which allows for different ways for the non-local representation.
Following a mathematical description of non-locality [Rogula, 1982], an abstract form of the fundamental equations governing any physical theory can be expressed as
Au=f , (1.36)
whereby f is a given excitation, u denotes the unknown response, and A is the corre-
sponding operator which characterizes the system and may possibly be non-linear. Since Acharacterizes the system, it also determines its locality properties. ForAto belocal, the following property must be satisfied:
Definition 1. If two functions u and v are identical in an open set O, then their images Au andAvare also identical inO.
In other words, whenever the identityu(x) =v(x) holds for all pointsxin a neighborhood ofx0, then it also follows that Au(x0) =Av(x0). This condition is satisfied for differential operators, and we may hence define local theories as those being fully described by differ- ential equations. Non-local theories, on the other hand, are based on integrodifferential equations.
However, this definition of locality is of a somewhat narrow nature, and a different de- scription relating to the absence or presence of a characteristic length can be given. In theories in which a characteristic length is absent, the fundamental equations are in- variant under scaling of the spatial coordinates [Rogula, 1982]. Local theories satisfying this property are denoted strictly local, whereas local theories not invariant under spa- tial scaling are calledweakly non-local. Typically, theories of the weakly non-local group contain differential equations with derivatives of different orders. By taking the ratio of coefficients multiplying these terms of different order (which have different physical dimensions), it is then possible to find a characteristic length.
As a simple example of weakly non-local theories, we may look at a Timoshenko beam, which relates to the previously mentioned Cosserat continuum as a specific one-dimensional version. In this case, the characteristic length is given by the ratio of the square roots of the bending and shear stiffness values of a cross-section. Under the assumption of a fixed cross-sectional shape, it follows that the characteristic length is proportional to the depth of a Timoshenko beam. Hence, only the beam span remains as an actual geomet- ric dimension of the model, whereas the depth is described by means of a generalized
material model and given in terms of moment–curvature (or shear force to shear distor- tion) relations. Therefore, in this example the material length scale can be traced back to a geometric dimension that is not explicitly resolved anymore, viz. the beam’s depth.
Similarly, characteristic lengths in generalized continuum models are the result of a ho- mogenization procedure; they thus represent characteristics of the heterogeneity, which are not explicitly resolved anymore.
In summary, continuum models may be classified according to:
• strictly local models (non-polar simple materials),
• weakly non-local models (polar and gradient theories),
• strongly non-local materials (models of integral type).
1.3.3. Non-local models of integral type
Non-local elasticity
As introduced in [Edelen and Laws, 1971; Eringen, 1972], non-local elasticity theories involve many different fields of non-local character (e.g. body forces, mass or internal energy), which made their application to real problems a challenging task. In further de- velopments, theories of non-locality were reduced to only include a non-local character in their stress-strain relations, while keeping the local character of equilibrium and kine- matic equations, as well as of the corresponding boundary conditions [Eringen and Kim, 1974]. More recently, a variational model has been developed [Polizzotto, 2001], which introduces the quadratic energy functional
W =1 2
Z
V
Z
V
ε(x)Cl(x,ζ)ε(x)dxdζ (1.37)
under the assumptions of small strains and linear elasticity. Here,ε(x) denotes the strain field and Cl(x,ζ) is the elastic stiffness in a generalized form. In this form, locality is recovered for
Cl(x,ζ) =Cl
![Figure 1.1.: Young’s modulus versus mass density plot for different groups of materials, adapted from [Granta Design, 2014].](https://thumb-ap.123doks.com/thumbv2/123dok/12145729.0/16.918.292.626.145.405/figure-modulus-density-different-materials-adapted-granta-design.webp)
![Figure 1.4.: Crazing process in a steel/polyurea/steel sandwich specimen under opening mode fracture [Yong et al., 2009].](https://thumb-ap.123doks.com/thumbv2/123dok/12145729.0/19.918.290.631.142.304/figure-crazing-process-polyurea-sandwich-specimen-opening-fracture.webp)
![Figure 1.6.: Representative volume element used in the eight-chain-model [Arruda and Boyce, 1993].](https://thumb-ap.123doks.com/thumbv2/123dok/12145729.0/27.918.214.710.146.332/figure-representative-volume-element-chain-model-arruda-boyce.webp)
![Figure 1.7.: Network of cross-linked polymer chains upon deformation (adapted from Weiner [2002]).](https://thumb-ap.123doks.com/thumbv2/123dok/12145729.0/32.918.250.669.138.277/figure-network-linked-polymer-chains-deformation-adapted-weiner.webp)
